268 ANALYTICAL MECHANICS
Therefore the resultant external torque acting upon the body is
or
where / is supposed to vary with the time. Equation (III) is the general form of torque equation, of which equation (V) of Chapter XI is a special case.
Introducing this expression of G in the definition for angular impulse we obtain
H= r
Jo
-JT
= Jco - I0a>0, (IV)
where 70 and co0 denote the moment of inertia and the angular acceleration at the instant t = 0, and I and co those at t = t. Equation (IV) is a generalization of equation (II). It states that angular impulse equals the change in the angular momentum under all circumstances.
211. The Principle of the Conservation of Angular Momentum.— When the resultant external torque acting upon a body or system of bodies vanishes, it follows from equation (III) that
and consequently /co = const. (V)
Therefore if the resultant of all the external torques acting upon a system vanishes, the angular momentum of the system